极限问题

wangyangkee

这---------知道了，你不懂这里谈的是序列的极限--------句话，不健康；

wangyangkee

不管如何考虑，
楼主的函数始终存在极值1和极值0；
函数是连续的；
极限不存在，，，
-----------------------------------显然对任何正整数 n, |sin n| > 0, 由于 π是无理数而 |sin x| 以 π 为周期，存在自然数的子序列 {n(k)} 使得 {|sin (n(k))|} 的极限为0。由17楼，可见
lim |sin (n(k))| = 0   然而
k →∞
lim |sin (n(k))|^(1/n(k)) = 1   
k →∞
直观地说， n(k) 靠近 π 的整数倍的程度， 即 |sin (n(k))| 接近于0 的程度不够，以至于对 |sin (n(k))| 取 n(k) 次根后会逼近 1.
取 ε∈ (0,1), 我们知道存在正整数 N(ε), 使得 n > N(ε) 时恒有
|sin n|> (1-ε)^n, 这就是无规则的序列{|sin n|}背后的规则.
有兴趣的话可以算算 {|sin n|} 的前 1000 项， 看看有多乱！ 注意 sin n 中的 n 是按弧度参与计算的。---------------------------------------------------
  

elimqiu

[这个贴子最后由elimqiu在 2010/08/29 00:23am 第 2 次编辑]

序列 {|sin n|} 的前1000项, 取前五位小数
0.84147  0.90930  0.14112  0.75680  0.95892  0.27942  0.65699  0.98936  0.41212  0.54402
0.99999  0.53657  0.42017  0.99061  0.65029  0.28790  0.96140  0.75099  0.14988  0.91295
0.83666  0.00885  0.84622  0.90558  0.13235  0.76256  0.95638  0.27091  0.66363  0.98803
0.40404  0.55143  0.99991  0.52908  0.42818  0.99178  0.64354  0.29637  0.96380  0.74511
0.15862  0.91652  0.83177  0.01770  0.85090  0.90179  0.12357  0.76825  0.95375  0.26237
0.67023  0.98663  0.39593  0.55879  0.99976  0.52155  0.43616  0.99287  0.63674  0.30481
0.96612  0.73918  0.16736  0.92003  0.82683  0.02655  0.85552  0.89793  0.11478  0.77389
0.95105  0.25382  0.67677  0.98515  0.38778  0.56611  0.99952  0.51398  0.44411  0.99389
0.62989  0.31323  0.96836  0.73319  0.17608  0.92346  0.82182  0.03540  0.86007  0.89400
0.10599  0.77947  0.94828  0.24525  0.68326  0.98359  0.37961  0.57338  0.99921  0.50637
0.45203  0.99483  0.62299  0.32162  0.97054  0.72714  0.18478  0.92682  0.81674  0.04424
0.86455  0.89000  0.09718  0.78498  0.94544  0.23666  0.68970  0.98195  0.37140  0.58061
0.99882  0.49871  0.45990  0.99569  0.61604  0.32999  0.97263  0.72104  0.19347  0.93011
0.81160  0.05308  0.86897  0.88592  0.08837  0.79043  0.94251  0.22805  0.69608  0.98024
0.36317  0.58780  0.99835  0.49102  0.46775  0.99647  0.60904  0.33833  0.97465  0.71488
0.20215  0.93332  0.80640  0.06192  0.87331  0.88178  0.07955  0.79582  0.93952  0.21943
0.70241  0.97845  0.35491  0.59493  0.99780  0.48329  0.47555  0.99717  0.60200  0.34665
0.97659  0.70866  0.21081  0.93646  0.80113  0.07075  0.87759  0.87758  0.07072  0.80115
0.93645  0.21078  0.70868  0.97658  0.34662  0.60202  0.99717  0.47552  0.48332  0.99780
0.59491  0.35494  0.97846  0.70239  0.21945  0.93953  0.79581  0.07958  0.88180  0.87330
0.06189  0.80642  0.93331  0.20212  0.71490  0.97464  0.33831  0.60907  0.99647  0.46772
0.49105  0.99835  0.58777  0.36320  0.98025  0.69606  0.22808  0.94252  0.79041  0.08840
0.88594  0.86895  0.05305  0.81162  0.93009  0.19344  0.72106  0.97262  0.32996  0.61606
0.99568  0.45988  0.49874  0.99882  0.58059  0.37143  0.98196  0.68968  0.23669  0.94545
0.78496  0.09721  0.89001  0.86454  0.04421  0.81676  0.92681  0.18475  0.72716  0.97053
0.32159  0.62301  0.99482  0.45200  0.50639  0.99921  0.57336  0.37964  0.98359  0.68324
0.24528  0.94829  0.77945  0.10602  0.89401  0.86005  0.03537  0.82184  0.92345  0.17605
0.73321  0.96836  0.31320  0.62991  0.99389  0.44409  0.51400  0.99952  0.56608  0.38781
0.98515  0.67675  0.25385  0.95106  0.77387  0.11481  0.89794  0.85550  0.02652  0.82685
0.92001  0.16733  0.73920  0.96611  0.30478  0.63676  0.99287  0.43614  0.52158  0.99976
0.55876  0.39595  0.98663  0.67021  0.26240  0.95376  0.76824  0.12360  0.90180  0.85089
0.01767  0.83179  0.91651  0.15859  0.74513  0.96379  0.29634  0.64356  0.99177  0.42816
0.52911  0.99991  0.55140  0.40407  0.98804  0.66361  0.27093  0.95638  0.76254  0.13238
0.90559  0.84620  0.00882  0.83667  0.91293  0.14985  0.75101  0.96139  0.28787  0.65031
0.99060  0.42014  0.53660  0.99999  0.54400  0.41215  0.98936  0.65696  0.27944  0.95893
0.75678  0.14115  0.90931  0.84145  0.00003  0.84149  0.90928  0.14109  0.75682  0.95892
0.27939  0.65701  0.98935  0.41209  0.54405  0.99999  0.53655  0.42019  0.99061  0.65026
0.28793  0.96141  0.75097  0.14991  0.91296  0.83664  0.00888  0.84624  0.90557  0.13232
0.76258  0.95637  0.27088  0.66366  0.98803  0.40401  0.55145  0.99991  0.52906  0.42821
0.99178  0.64352  0.29640  0.96380  0.74509  0.15865  0.91653  0.83176  0.01773  0.85092
0.90178  0.12354  0.76827  0.95374  0.26235  0.67025  0.98662  0.39590  0.55881  0.99975
0.52153  0.43619  0.99288  0.63671  0.30484  0.96613  0.73916  0.16739  0.92004  0.82681
0.02658  0.85554  0.89791  0.11475  0.77391  0.95105  0.25379  0.67679  0.98514  0.38775
0.56613  0.99952  0.51395  0.44414  0.99389  0.62986  0.31326  0.96837  0.73317  0.17611
0.92347  0.82180  0.03543  0.86008  0.89398  0.10596  0.77948  0.94827  0.24522  0.68328
0.98358  0.37958  0.57341  0.99921  0.50634  0.45205  0.99483  0.62297  0.32165  0.97054
0.72712  0.18481  0.92683  0.81673  0.04427  0.86457  0.88998  0.09715  0.78500  0.94543
0.23663  0.68972  0.98195  0.37138  0.58064  0.99881  0.49869  0.45993  0.99569  0.61602
0.33002  0.97264  0.72102  0.19350  0.93012  0.81159  0.05311  0.86898  0.88591  0.08834
0.79045  0.94250  0.22802  0.69610  0.98023  0.36314  0.58782  0.99834  0.49100  0.46777
0.99647  0.60902  0.33836  0.97466  0.71486  0.20218  0.93333  0.80638  0.06195  0.87333
0.88177  0.07952  0.79584  0.93951  0.21940  0.70243  0.97844  0.35488  0.59496  0.99780
0.48327  0.47558  0.99718  0.60198  0.34668  0.97660  0.70864  0.21084  0.93647  0.80112
0.07078  0.87760  0.87756  0.07069  0.80117  0.93644  0.21075  0.70870  0.97658  0.34659
0.60205  0.99717  0.47550  0.48334  0.99780  0.59488  0.35497  0.97846  0.70236  0.21948
0.93954  0.79579  0.07961  0.88181  0.87328  0.06186  0.80644  0.93330  0.20209  0.71492
0.97464  0.33828  0.60909  0.99646  0.46769  0.49107  0.99835  0.58775  0.36323  0.98025
0.69604  0.22811  0.94253  0.79040  0.08843  0.88595  0.86894  0.05302  0.81164  0.93008
0.19341  0.72108  0.97262  0.32993  0.61609  0.99568  0.45985  0.49877  0.99882  0.58056
0.37146  0.98196  0.68965  0.23672  0.94545  0.78494  0.09724  0.89002  0.86452  0.04418
0.81678  0.92680  0.18472  0.72718  0.97052  0.32157  0.62304  0.99482  0.45197  0.50642
0.99921  0.57333  0.37966  0.98360  0.68322  0.24531  0.94830  0.77943  0.10605  0.89402
0.86004  0.03534  0.82185  0.92344  0.17602  0.73323  0.96835  0.31317  0.62993  0.99388
0.44406  0.51403  0.99952  0.56606  0.38784  0.98516  0.67673  0.25388  0.95107  0.77385
0.11484  0.89795  0.85549  0.02649  0.82686  0.92000  0.16730  0.73922  0.96610  0.30475
0.63678  0.99287  0.43611  0.52160  0.99976  0.55874  0.39598  0.98664  0.67018  0.26243
0.95377  0.76822  0.12363  0.90181  0.85087  0.01764  0.83181  0.91650  0.15856  0.74515
0.96378  0.29631  0.64358  0.99177  0.42813  0.52913  0.99991  0.55138  0.40409  0.98804
0.66359  0.27096  0.95639  0.76252  0.13241  0.90560  0.84619  0.00879  0.83669  0.91292
0.14982  0.75103  0.96138  0.28785  0.65033  0.99060  0.42011  0.53662  0.99999  0.54397
0.41217  0.98937  0.65694  0.27947  0.95894  0.75676  0.14118  0.90932  0.84144  0.00006
0.84150  0.90927  0.14106  0.75684  0.95891  0.27936  0.65703  0.98935  0.41206  0.54407
0.99999  0.53652  0.42022  0.99062  0.65024  0.28796  0.96141  0.75095  0.14994  0.91297
0.83662  0.00891  0.84625  0.90555  0.13229  0.76260  0.95636  0.27085  0.66368  0.98802
0.40398  0.55148  0.99991  0.52903  0.42824  0.99179  0.64349  0.29643  0.96381  0.74507
0.15868  0.91655  0.83174  0.01776  0.85094  0.90176  0.12351  0.76829  0.95373  0.26232
0.67027  0.98662  0.39587  0.55884  0.99975  0.52150  0.43622  0.99288  0.63669  0.30487
0.96613  0.73914  0.16742  0.92005  0.82679  0.02661  0.85555  0.89790  0.11472  0.77393
0.95104  0.25377  0.67682  0.98514  0.38773  0.56616  0.99952  0.51393  0.44417  0.99390
0.62984  0.31329  0.96838  0.73315  0.17613  0.92348  0.82178  0.03546  0.86010  0.89397
0.10593  0.77950  0.94826  0.24519  0.68331  0.98358  0.37955  0.57343  0.99920  0.50631
0.45208  0.99483  0.62294  0.32168  0.97055  0.72710  0.18484  0.92684  0.81671  0.04430
0.86458  0.88997  0.09712  0.78502  0.94542  0.23660  0.68974  0.98194  0.37135  0.58066
0.99881  0.49866  0.45996  0.99569  0.61599  0.33005  0.97264  0.72100  0.19353  0.93013
0.81157  0.05314  0.86900  0.88590  0.08831  0.79047  0.94249  0.22799  0.69612  0.98023
0.36312  0.58784  0.99834  0.49097  0.46780  0.99647  0.60900  0.33839  0.97466  0.71483
0.20221  0.93334  0.80636  0.06198  0.87334  0.88176  0.07949  0.79586  0.93950  0.21937
0.70245  0.97844  0.35485  0.59498  0.99779  0.48324  0.47560  0.99718  0.60195  0.34671
0.97660  0.70862  0.21087  0.93648  0.80110  0.07081  0.87762  0.87755  0.07066  0.80119
0.93643  0.21072  0.70872  0.97657  0.34656  0.60207  0.99717  0.47547  0.48337  0.99780
0.59486  0.35499  0.97847  0.70234  0.21951  0.93955  0.79577  0.07964  0.88183  0.87327
0.06183  0.80645  0.93329  0.20206  0.71494  0.97463  0.33825  0.60912  0.99646  0.46767
0.49110  0.99835  0.58772  0.36326  0.98026  0.69602  0.22814  0.94254  0.79038  0.08846
0.88597  0.86892  0.05299  0.81166  0.93007  0.19338  0.72110  0.97261  0.32991  0.61611
0.99568  0.45982  0.49879  0.99882  0.58054  0.37149  0.98197  0.68963  0.23675  0.94546
0.78492  0.09727  0.89004  0.86451  0.04415  0.81679  0.92678  0.18469  0.72720  0.97051
0.32154  0.62306  0.99482  0.45195  0.50644  0.99921  0.57331  0.37969  0.98360  0.68320
0.24534  0.94831  0.77941  0.10608  0.89404  0.86002  0.03531  0.82187  0.92342  0.17599
0.73325  0.96834  0.31314  0.62996  0.99388  0.44403  0.51406  0.99952  0.56603  0.38786
0.98516  0.67671  0.25391  0.95108  0.77383  0.11487  0.89797  0.85547  0.02646  0.82688

wangyangkee

elimqiu 上面的演示，没有涉及无理数，，，

elimqiu

楼上的大概没见过数学手册里的三角函数表？ 哈哈，有理数/无理数？

wangyangkee

下面引用由elimqiu在 2010/08/29 07:25am 发表的内容：
楼上的大概没见过数学手册里的三角函数表？ 哈哈，有理数/无理数？

-------------楼上的大概没见过数学手册里的三角函数表？ 哈哈，有理数/无理数？---------------------
  

wangyangkee

下面引用由elimqiu在 2010/08/29 06:52am 发表的内容： 序列 {|sin n|} 的前1000项, 取前五位小数
0.84147  0.90930  0.14112  0.75680  0.95892  0.27942  0.65699  0.98936  0.41212  0.54402
0.99999   ...

-------------序列 {|sin n|} 的前1000项, 取前五位小数---------------

elimqiu

[这个贴子最后由elimqiu在 2010/08/29 03:49am 第 2 次编辑]

如果把[0,1) 20 等分， 那么前n项落在各小区间 [i/20, (i+1)/20) 的计数有下列分布
In [51]: sta(200)
[6, 8, 6, 5, 8, 6, 7, 7, 6, 9, 6, 8, 8, 8, 10, 10, 11, 14, 17, 40]
In [52]: sta(1000)
[31, 36, 32, 28, 36, 31, 35, 35, 31, 42, 32, 40, 40, 44, 46, 50, 55, 71, 84, 201]
In [53]: sta(2000)
[61, 71, 65, 56, 70, 66, 68, 68, 66, 81, 66, 79, 79, 89, 91, 101, 113, 137, 169, 404]
In [54]: sta(3000)
[94, 101, 98, 89, 100, 102, 101, 101, 102, 117, 102, 118, 118, 136, 134, 153, 169, 203, 254, 608]
In [55]: sta(30000)
[930,1015,934,937,1002,1014,1014,1014,1023,1139,1077,1156,1254,1280,1387,1508,1696,2001,2535,6084][br][br][color=#990000]-=-=-=-=- 以下内容由 elimqiu 在 时添加 -=-=-=-=-
这种“右倾”现象如何解释？ 十分有趣！
从单位圆上的均匀分布到纵轴上投影的非均匀分布，我们似乎可以理解非均匀分布。但具体的分析论证还是不可少的。

luyuanhong

[这个贴子最后由luyuanhong在 2010/08/29 03:22pm 第 1 次编辑]

下面引用由elimqiu在 2010/08/29 00:57am 发表的内容：
如果把[0,1) 20 等分， 那么前n项落在各小区间 [i/20, (i+1)/20) 的计数有下列分布
In [51]: sta(200)
[6, 8, 6, 5, 8, 6, 7, 7, 6, 9, 6, 8, 8, 8, 10, 10, 11, 14, 17, 40]
In [52]: sta(1000)
[31, 36, 32, 28, 36, 31, 35, 35, 31, 42, 32, 40, 40, 44, 46, 50, 55, 71, 84, 201]
In [53]: sta(2000)
[61, 71, 65, 56, 70, 66, 68, 68, 66, 81, 66, 79, 79, 89, 91, 101, 113, 137, 169, 404]
In [54]: sta(3000)
[94, 101, 98, 89, 100, 102, 101, 101, 102, 117, 102, 118, 118, 136, 134, 153, 169, 203, 254, 608]
In [55]: sta(30000)
[930,1015,934,937,1002,1014,1014,1014,1023,1139,1077,1156,1254,1280,1387,1508,1696,2001,2535,6084]
这种“右倾”现象如何解释？ 十分有趣！
从单位圆上的均匀分布到纵轴上投影的非均匀分布，我们似乎可以理解非均匀分布。但具体的分析论证还是不可少的。

当 x 在数轴上均匀分布时，sinx 的取值就会发生这种“右倾”现象。
从下面的表格可以看出，观测到的频数，与按照 x 均匀分布计算出来的结果，符合得很好：

elimqiu

对！可以把均匀分布在单位圆上的点投影到 Y 轴上来理解这种倾向。[br][br]-=-=-=-=- 以下内容由 elimqiu 在 时添加 -=-=-=-=-
谢谢路老师的计算